On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

Abstract

We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus g≥1 and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a power deviation spectrum and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni [Ann. of Math. (2) 155 (2002), 1-103] and Bufetov [Ann. of Math. (2) 179 (2014), 431-499] and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable correction operators for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author [Ann. of Math. (2) 173 (2011), 1743-1778] we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus g≥2.